All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
It is Islamically permissible to mention the good traits and good deeds of the deceased, as this is a testimony in his favor. Anas said: “A funeral procession passed by and the people praised the deceased, so the Prophet said: "It has become certain.” Then another funeral procession passed by and the people spoke badly of the deceased. The Prophet said: “It has become certain.” ‘Umar asked the Prophet: “What do you mean by 'It has become certain?'” He replied: "You praised this [person], so Paradise has become certain for him; and you spoke badly of this [person], so Hellfire has become certain for him. You people are Allaah’s witnesses on earth.” [Al-Bukhari and Muslim]
On the other hand, if the parents mention that they gave their children a good upbringing, if this is for a correct intentions, like speaking about the favors of Allaah or urging others to do so, then this is permissible. However, doing so is Islamically forbidden if the intention is showing off or boasting, for example.
It should be mentioned that the Hadeeth: “Mention the good deeds of your deceased people...” is a Dha’eef [weak] Hadeeth. Shaykh Al-Albaani stated that it was weak in his revision of Sunan Abu Daawood and Sunan At-Tirmithi, as well as in Dha’eef Al-Jaami’ As-Sagheer and in Dha’eef At-Targheeb.
At any rate, whoever dies upon Islam (i.e. as a Muslim), it is permissible to ask Allaah to have mercy upon him and forgive him and not to mention his sins, because as he has reached the result of what he forwarded (i.e. the Hereafter, the place of reward or punishment for his deeds). Who knows, maybe Allaah has forgiven him his bad deeds and sins; as per the Hadeeth which reads: “Do not insult the dead as they have reached the result of what they forwarded [i.e. the Hereafter].” [Al-Bukhari]
In addition, speaking ill of the dead hurts the living, and it is prohibited to harm others. Hence, it is mentioned in a Hadeeth: “Do not insult our dead, because if you do so, you harm the living.” [Al-Haakim and Ath-Thahabi]
Allaah Knows best.